An equivalent condition for a uniform space to be coverable

An equiv alen t condition f or a unif orm space t o b e co verable Conrad Plaut Mathematics Departmen t Univ ersit y o f T ennesse e Ayres Hall 121 Kno xville, TN 379 96-1300 cplaut@math.utk.edu No v em b er 21 , 2018 Abstract W e prov e that an equiv alent condition for a uniform space to b e cov- erable is that the images of the natural pro jections in the fundamental inv erse system are uniformly op en in a certain sense. As corollaries w e (1) obtain a concrete wa y to fi nd co vering en tourage, (2) correct an error in [3], and (3) show th at cov erable is equiva lent to chain connected and uniformly joinable in the sense of [5]. Keywords: univ ersal cov er, uniform space, cov erable, fund amen tal group MSC: 55Q52; 54E15,55M10 In [6] and [3] w e approached the pro blem of defining universal cov er s o f lo cally bad s paces using the following ideas: (1) The a ppropriate ca tegory in which to w o rk in is not top olog ical spa c e s, but rather uniform spa ces and uni- formly contin uous mappings. (2) The replacement for quotient mapping s in this category is bi-unifo r mly contin uous mappings (s e e b elow). (3) T he appropriate replacement for curves is equiv alence classes of chains. W e sho wed in [3] that such a progr a m can b e car ried out for a lar ge class of uniform s paces called c over able sp ac es. In pa rticular we constructed, for any uniform spac e X , a uni- form spa ce e X , a natural uniformly cont inuous mapping φ : e X → X and a group δ 1 ( X ) that acts on e X . F or cov erable spaces the ma pping φ has ma n y o f the prop erties of a universal covering map, such as lifting and universal prop erties, and w e refer to the space e X a s the uniform univ ersa l co ver of X . The gro up δ 1 ( X ) (which we called the “deck group” in [3] but whic h was renamed the “uni- form fundamen tal group” in [7]) is a functoria l inv ar iant o f unifor m structures having pr op erties like the fundamental gr oup in this categor y . See [3] for many sp ecific theorems and examples, and [4] fo r some a dditional applications. The space e X is the in verse limit of the fundamental in verse s ystem ( X E , φ E F ) of X , which is indexed on the s e t of a ll entourages E of X . Roug hly sp eaking , X E “unrolls” nontrivial cla sses of lo ops that are in some sense larg er than E 1 (w e will give more ba ckground b elow). W e will denote by φ E : e X → X E and φ : e X → X the natural pro jections (the latter is a ctually just the endp oint mapping). X is coverable by definition if φ is sur jectiv e φ E is s urjective for all E in s ome ba sis (called a cov er ing basis) for the uniform str ucture of X . Elements of the cov ering ba sis are called c ov ering ent o urages. Existence o f a cov ering basis ca n b e prov ed in many cas es–for example it is not to o difficult to show that connected and uniformly lo cally connected pseudo metric spaces– whic h includes all geo desic spa c e s–are cov erable (Theorem 98 , [3]). On the other hand, it is easy to find in cov erable spa ces en tour a ges that are not cov ering entourages (cf. Example 16), a nd without uniform nice lo ca l c o nditions it can be difficult to verify c overabilit y . In this pap er we show that cov erability is eq uiv alent to the following: X is chain connected a nd for any entourage E , φ E has ima ge that is uniformly op en in X E in a sense defined below (Theo rem 1 2). While surjectivity of maps in an inv erse sys tem is g enerally a strong condition useful for pro ving theorems, fro m the standp oint of verification there is a clear adv antage in not having to hun t for the covering entourages. Theore m 12 provides a condition that may be verified for arbitr ary en toura ges. Moreov er, a s a coro lla ry we obtain a constr uctive metho d for extr acting a covering entourage fr o m an arbitra ry ent o urage without having to consider the mapping φ E at all (see Corollary 15 and Example 1 6). In [5] the authors explor e our construction o f the uniform universal cov er in the setting of what they call uniformly joinab le uniform spaces. A t first their construction do es not lo ok like our construction, and o ne must lo o k in Section 8 of [5] to find a s tatemen t that they a re “identical”. As we explain below, their definitions of GP ( X, ∗ ) a nd ∨ π ( X , ∗ ) are simply tr a nslations of our definitions of e X and δ 1 ( X ) into the language of Rips co mplexes. Moreov er , another applica- tion of our main theor em is that the cla ss of chain co nnec ted, uniformly joinable spaces considered in [5] is precisely the same as the cla ss of coverable s paces (Corollar y 17). The authors o f [5] state that “a topo logist would b e skeptical” of this par ticular r esult–an asse r tion back ed by some musings on Sieb enmann’s thesis. Nonetheless, their definition is clo sely related to concepts in contin ua theory and they obtain the int er esting result that a metric co mpactum X is uniformly joinable if a nd o nly if the function φ : e X → X is surjective. In light of Cor ollary 17, for metrizable spaces this is a strong genera lization of the fac t, prov ed in [2], that a compact top ologica l gr oup is coverable if a nd only if φ is surjective. Also in [5] the author s intro duce a notion of gener alized cov er in the uniform category that do es not require a group action. In light of this, wha t we called “cov ers of uniform spaces” in [3] really should be ca lled something like “regular uniform cov ers” a s is suggested in [5 ]. W e do not use any theor ems from [5] in our pro ofs and in fact, in lig h t of Corollar y 17, some of the theorems in [5] were already proved in [3]. Thank s to Jurek Dydak fo r critiques and s timulating comments. In par ticular, he po in ted out an e r ror in [3] that is cor rected in the pr esent pap er. V alera Berestovskii provided some v aluable comments. W e will use the notation of [3]. In particular , we gener ally use f in plac e of 2 f × f ; for example, if E is an entourage in a uniform space we w ill write f ( E ) rather than ( f × f ) ( E ). F or an en tour age E we let B ( x, E ) := { y : ( x, y ) ∈ E } . Given a uniform space X , for an y entourage E , X E is defined to b e the space of E -homo to p y classes o f E -chains α := { x 0 = ∗ , ..., x n } , where ∗ is a base- po in t. By definition, α is an E - c hain if ( x i , x i +1 ) ∈ E for all i . An E -homotopy of α is a finite sequence o f mov es s tarting with α , where ea ch mov e consists of a dding or taking aw ay a p oint (but not e ndpoints!) so long as doing so results in an E -chain. F or chain connected spaces (meaning every pair of po in ts is joined by an E -chain for all E ) nothing of conseq uence dep ends on the choice of basep oint so we gener a lly eliminate it fr om the notation. The space X E , the elements of which are denoted [ α ] E , has a natur al uniform struc- ture having a basis consisting of sets F ∗ , where F ⊂ E and ([ α ] E , [ β ] E ) ∈ F ∗ if and only if [ α ] E = [ x 0 , ..., x n − 1 , x ] E , and [ β ] E = [ x 0 , ..., x n − 1 , y ] E , with ( x, y ) ∈ F . e X is given the inv ers e limit unifor mit y . When F ⊂ E , the mapping φ E F : X F → X E simply c onsiders an F -chain a s a n E -chain, i.e., φ E F ([ α ] F ) = [ α ] E , and φ X E : X E → X is the endpoint mapping. With res pect to the natural uniform structure these mappings are bi-unifor mly co n tinuous in the sense that the inverse image of an y en toura ge is an en toura ge, a nd the image of any en toura ge is an en toura ge in the subspace uniformit y of the im- age of the mapping. Given a unifor mly contin uous mapping f : X → Y and ent o urages E , F in X , Y , r e spec tively , such that f ( E ) ⊂ F , there is a unique basep oint-preserving induced uniformly contin uous function f E F : X E → Y F such that φ Y F ◦ f E F = f ◦ φ X E , which simply takes [ α ] E to [ f ( α )] F . If X is chain connected, the function φ X E : X E → X is a q uotient ma pping via the isomor phic action of the group δ E ( X ) consisting of E -homotopy cla sses of E -lo ops. Pr ecisely what this means is not needed for this pap er (see [6] for the definitions); we can get b y with t wo facts: first, if φ X E ( a ) = φ X E ( b ) then for some g ∈ δ E ( X ), g ( a ) = b and seco nd, the en toura g es F ∗ are invariant in the sense that for every g ∈ δ E ( X ), g ( F ∗ ) = F ∗ . Definition 1 We say that a subset A of a un iform sp ac e X is u niformly op en if ther e is an entour age E in X such that for every a ∈ A , B ( a, E ) ⊂ A . There are a few o bvious fa c ts: if A is a uniformly op en set then A is op en, the complement of A is uniformly op en, a nd hence A its e lf is a lso closed. But for example in the r ational num b ers Q with the usual metric there are plen ty of op en a nd closed subse ts that ar e not uniformly open. The inv erse imag e of any unifor mly o pen set via a uniformly contin uous function is uniformly op en, but in genera l nothing can b e sa id of ima ges. F or e x ample co nsider the bi- uniformly contin uous s urjection f : [0 , 2] × Z 2 → [0 , 2] defined by ( q , 0) 7− → q and ( q , 1) 7− → q 2 . Here [0 , 2] has its usua l metr ic, Z 2 has the discr ete metric, and [0 , 2] × Z 2 → [0 , 2] has the pro duct metric. It is ea sy to c heck that [0 , 2] × { 1 } is uniformly open in [0 , 2] × Z 2 but of course f ([0 , 2] × { 1 } ) = [0 , 1] is not even op en in [0 , 2]. (But see Remark 6 b elow.) Lemma 2 A uniform sp ac e X is chain c onne cte d if and only if the only non- empty uniformly op en subset of X is X . 3 Pro of. Supp ose X is chain c o nnected a nd let U b e a nonempt y uniformly open subset o f X . If E is an entourage as in the definition of uniformly o pen, then any E - c ha in starting at x canno t leav e U and so U = X . F or the conv er se, let x ∈ X , E b e an entourage, and U b e the se t of a ll p oints that a r e joined to x by an E -c hain. If z ∈ U then clearly B ( z , E ) ∈ U ; hence U is uniformly open and non-empty , hence equa l to X . Since E and x were arbitrary , X is chain connected. Obviously the intersection of any tw o uniformly op e n subsets is uniformly op en. As a coro llary of this and the ab ov e lemma we obtain: Corollary 3 If any two uniformly op en chain c onne cte d subsets of a uniform sp ac e X have non-empty interse ction then they must b e e qual. Definition 4 If X is a uniform sp ac e, E ⊂ F ar e entour ages in X , and A is a uniformly op en subset of X E , define F A := φ X E ( F ∗ ∩ ( A × A ) ) . Lemma 5 L et X b e a chain c onne cte d un iform sp ac e, E b e an entour age in X , and A b e a uniformly op en subset of X E . Then φ X E ( A ) = X and for any entour age F ⊂ E , F A is an entour age in X . Pro of. Consider en tour ages W ⊂ F ⊂ E such that if x ∈ A and ( x, y ) ∈ W ∗ then y ∈ A . W e will first prove that φ X E ( A ) is uniformly o p en a nd he nc e equal to X . Suppose that ( a, b ) ∈ W and a ∈ φ X E ( A ). So there exist z ∈ A such that φ X E ( z ) = a and since W = φ X E ( W ∗ ) (cf. [3], P rop osition 16 ) ther e exists ( x, y ) ∈ W ∗ such that φ X E ( x, y ) = ( a, b ). Next there exists so me g ∈ δ E ( X ) such that g ( x ) = z . By the inv ariance of W ∗ , if w := g ( y ) then ( z , w ) ∈ W ∗ and φ X E ( z , w ) = ( a, b ). By choice of W ∗ , w ∈ A , which pla ces b ∈ φ X E ( A ), finishing the pro of that φ X E ( A ) is uniformly o pen and equal to X . Now the initial assumption that a ∈ φ X E ( A ) is s up erfluo us and the same arg umen t shows W ⊂ W A and since W A ⊂ F A , F A is an entourage. Remark 6 The same pr o of as in the pr evious lemma shows the fol lowing: If f : X → Y is a quotient of uniform sp ac es via an isomorphic action and A ⊂ X is u niformly op en then f ( A ) is uniformly op en in Y (se e [6] for a discussion of isomorphi c actions). Corollary 7 If X is a chain c onne cte d u n iform sp ac e and t her e is some en- tour age E such that φ E ( e X ) is uniformly op en in X E then φ : e X → X is surje c- tive. Lemma 8 L et X b e a chain c onne cte d uniform sp ac e and E b e an entour age in X such that A := φ E ( e X ) is uniformly op en. If α := {∗ = x 0 , ..., x n } is an E A -chain then [ α ] E ∈ A . Pro of. W e will show b y induction that [ x 0 , ..., x k ] E ∈ A for all k ≤ n . Cer- tainly the statement is true for k = 0. Supp ose tha t [ x 0 , ..., x k ] E ∈ A . Now 4 ( x k , x k +1 ) ∈ E A and by definition there exist E -chains γ := { ∗ = y 0 , ..., y m , x k } and ω := { y 0 , ..., y m , x k +1 } such that [ γ ] E , [ ω ] E ∈ A a nd ( x k , x k +1 ) ∈ E . So we have ([ a D ] D ) , ([ b D ] D ) , ([ c D ] D ) ∈ e X such that [ a E ] E = [ x 0 , ..., x k ] E , [ b E ] E = [ γ ] E , and [ c E ] E = [ ω ] E . Consider κ := ([ a D ∗ b − 1 D ∗ c D ] D ) ∈ e X , where “ ∗ ” denotes concatenation of chains. Now φ E ( κ ) = [ a E ∗ b − 1 E ∗ c E ] E = [ { x 0 , ..., x k } ∗ γ − 1 ∗ ω ] E = [ x 0 , ..., x k , y m , ..., y 1 , y 0 , y 1 , ..., y m , x k +1 ] E = [ x 0 , ..., x k +1 ] E where the la st E - homotopy successively removes y 0 , y 1 , y 1 , y 2 , ..., y m . Prop osition 9 L et X b e a chain c onne cte d uniform sp ac e and E b e an en- tour age in X su ch t hat A := φ E ( e X ) is u niformly op en. L et ting D := E ∗ ∩ ( A × A ) and G := φ − 1 E ( D ) = φ − 1 E ( E ∗ ) , ther e is a uniformly c ontinuous function ψ : e X → A D such that the fol lowing diagr am c ommutes e X G φ f X G − → e X ↓ θ ւ ψ ↓ φ E A D φ AD − → A (1) wher e θ = ( φ E ) GD is the mapping induc e d by φ E . Pro of. Firs t of a ll we recall what it mea ns for ([ α ] E , [ β ] E ) ∈ D : [ α ] E , [ β ] E ∈ A and [ α ] E = [ y 0 , ..., y m , x ] E and [ β ] E = [ y 0 , ..., y m , y ] E for some choice of y 0 , ..., y m , with ( x, y ) ∈ E . W e will define ψ := f ◦ φ E A , wher e f ([ x 0 , ..., x n ] E A ) = [[ x 0 ] E , [ x 0 , x 1 ] E , ..., [ x 0 , ..., x n ] E ] D (2) W e need to chec k v a r ious things ab out f . First of a ll, note tha t b y Lemma 8 , [ x 0 , ..., x k ] E ∈ A for all 1 ≤ k ≤ n and therefore ([ x 0 , ..., x k ] E , [ x 0 , ..., x k , x k +1 ] E ) = ([ x 0 , ..., x k , x k ] E , [ x 0 , ..., x k , x k +1 ] E ) ∈ D so the definitio n at least go es int o the correct s et. T o see that f is well-defined, consider an E A -chain α ′ := { x 0 , ..., x k − 1 , x, x k , ..., x n } , which would lead to [[ x 0 ] E , ..., [ x 0 , ..., x k − 1 ] E , [ x 0 , ..., x k − 1 , x ] E , [ x 0 , ..., x k − 1 , x, x k ] E , ..., [ x 0 , ..., x k − 1 , x, x k , ..., x n ] E ] D in the above definition. But no tice that for any k ≤ m ≤ n , we a lready know { x 0 , ..., x k − 1 , x, x k , ..., x m } a nd { x 0 , ..., x k − 1 , x k , ..., x m } a r e E A -homotopic, he nce E -homotopic, so we ca n simply remov e the “ x ” fro m all suc h terms. This leav es the one extra term [ x 0 , ..., x k − 1 , x ] E . But s ince ([ x 0 , ..., x k − 1 ] E , [ x 0 , ..., x k − 1 , x k ] E ) ∈ D , up to D -homoto p y we may simply remov e this ter m, getting us back to (2). 5 W e will now check that f is unifor mly contin uous. T o do so we will hav e to be a little more ca r eful with notation. Given an en toura ge W ⊂ E A in X we hav e an ent our age called W ∗ in X E and one ca lled W ∗ in X E A . W e will refer to the latter as W # . W e also hav e the en toura ge ( W ∗ ∩ ( A × A )) ∗ in A D , which we will simply denote by W ∗∗ . The pro of o f uniform contin uity will b e finished if we c a n show that f ( W # ) ⊂ W ∗∗ . Let ([ α ] E A , [ β ] E A ) ∈ W # . By definition we may take α = { x 0 , ..., x n , x } and β = { x 0 , ..., x n , y } with ( x, y ) ∈ W . W e hav e f ([ α ] E A ) = [[ x 0 ] E , ..., [ x 0 , ..., x n ] E , [ x 0 , ..., x n , x ] E ] D f ([ β ] E A ) = [[ x 0 ] E , ..., [ x 0 , ..., x n ] E , [ x 0 , ..., x n , y ] E ] D Since ( x, y ) ∈ W a nd [ x 0 , ..., x n , x ] E , [ x 0 , ..., x n , y ] E ∈ A (Lemma 8 aga in), ([ x 0 , ..., x n , x ] E , [ x 0 , ..., x n , y ] E ) ∈ W ∗ ∩ ( A × A ) a nd ( f ([ α ] E A ) , f ([ β ] E A )) ∈ W ∗∗ . W e will now c heck the commutativit y of the diagr a m. Suppose that η := { y 0 = ∗ , y 1 , ..., y n } is a G -c ha in in e X . This mea ns that for all i , ( φ E ( y i ) , φ E ( y i +1 )) ∈ D . In particular, ( φ E ( y 0 ) , φ E ( y 1 )) = ([ ∗ ] E , φ E ( y 1 )) ∈ D . This means that we may write φ E ( y 1 ) = [ ∗ = w 0 , ..., w m , x 1 ] E , wher e x 1 is the endp oint of y 1 , { w 0 , ..., w m , ∗} is E - homotopic to the identit y a nd ( ∗ , x 1 ) ∈ E . But then we may use the null E -homo to p y of { w 0 , ..., w m , ∗} to see tha t φ E ( y 1 ) = [ ∗ , ..., w m , ∗ , x 1 ] E = [ ∗ , x 1 ] E . By definition of D we als o have [ ∗ , x 1 ] E = φ E ( y 1 ) ∈ A , which implies tha t {∗ , x 1 } is a n E A -chain. Pro ceeding inductively with essentially the same argument, we see tha t φ E ( y i ) = [ ∗ , x 1 , .., x i ] E , where x i is the endpoint of φ E ( y i ) and { x 0 , ..., x n } is an E A -chain. By definition of θ , θ ([ η ] G ) = [ φ E ( y 0 ) , ..., φ E ( y n )] D = [[ ∗ ] E , [ ∗ , x 1 ] E , ..., [ ∗ , x 1 , ..., x n ] E ] D = ψ ( y n ) = ψ ◦ φ e X G ([ η ] G ) This prov es the commutativit y of the upper tria ng le. The commutativit y of the low er triangle is o b vio us from the definition of ψ . Univ er sal unifor m spa ces a nd universal bases were defined in [3]; the defini- tions will b e explained in the pro of b elow. Prop osition 10 If X is a chain c onne cte d uniform sp ac e such that for every entour age E , φ E : e X → X E has uniformly op en image in X E then e X is universal with an invariant (with r esp e ct to the action of δ 1 ( X ) ) un iversal b asis. Pro of. Cons ider the diagr am (1). W e will sta rt b y s howing that e X is chain connected. Since φ E is surjective onto A , so is φ AD . This mea ns that every pair of p oints in A is joine d b y a D -chain. Equiv ale ntly , A × A = ∞ S n =1 D n (where D n is the set o f all po in ts in A joined to the basep oint by a D -chain of length n ). Since φ E is surjective, it is easy to check that φ − 1 E ( D n ) =  φ − 1 E ( D )  n =  φ − 1 E ( E ∗ )  n and so e X = ∞ S n =1  φ − 1 E ( E ∗ )  n (cf. the pro of of Lemma 1 1 in [3]). This means 6 that every pair of p oints in e X is joined by a φ − 1 E ( E ∗ )-chain. Since the set of all φ − 1 E ( E ∗ ) for ms a ba sis for the uniformity o f e X , e X is chain c o nnected. This now implies tha t the mapping φ e X G is s urjective and the hypothes e s of Prop osition 33 in [3] a re s atisfied for this diagra m, implying that φ e X G is a uniform homeomorphism. By definition G is a univ ersa l entourage, and since it is of the form φ − 1 E ( E ∗ ), it is inv ariant (cf. Pro po sition 41 of [6]). W e hav e shown that e X has a basis of univ ers al entourages; b y definition this makes e X universal. Alas, the pro o f of Corollar y 61 in [3] is not correct– or rather, the pro o f is correct for a weak er statement. The pe n ultimate sentence in the pro of r equires an additional assumpio n. F or exa mple, the pro of is cor rect for the following statement: Lemma 1 1 If f : X → Y is a quotient via an action on a uniform sp ac e X and X has a universal b asis that is invariant with r esp e ct to the action, then Y is c over able. Corollar y 61 was only used to establish the equiv alence of the definition of “cov era ble to polo gical group” as defined in [1] with the definition in [3] when applied to top ologica l groups co nsidered a s uniform spaces. In pa r ticular, no ne of the results of [1] cited in the current pap er relies o n this co rollary ; Lemma 11 will suffice to pr ov e our main Theorem 12, of which Coro llary 61 is a corollar y . Theorem 1 2 F or a chain c onne cte d uniform sp ac e X , the fol lowing ar e e quiv- alent: 1. X is c over able. 2. φ : e X → X is a bi-uniformly c ontinu ous surje ction. 3. F or e ach entour age E in X and any choic e of b asep oint, φ E ( e X ) is uni- formly op en in X E . Pro of. 1 ⇒ 2 follows from Theorem 4 5 in [3 ]. F or 2 ⇒ 3, let E be an ent o urage in X , A := φ E ( e X ). Supp ose that ([ α ] E , [ β ] E ) ∈ E ∗ A and [ α ] E ∈ A . So ther e is some ([ γ D ] D ) ∈ e X such tha t [ γ E ] E = [ α ] E and we may write α := { x 0 , ..., x n , x } and β := { x 0 , ..., x n , y } , with ( x, y ) ∈ E A . This in turn means that we have ([ α D ] D ) , ([ β D ] D ) ∈ e X with e ndp oints x and y such that ([ α E ] E ) , ([ β E ] E ) ∈ E ∗ ∩ ( A × A ). So we may now write α E := { y 0 , ..., y m , x } and β E := { y 0 , ..., y m , y } and ( x, y ) ∈ E . Consider ([ γ D ∗ α − 1 D ∗ β D ] D ) ∈ e X . Using an E -homo to p y like the one in the pro of o f Lemma 8 we have [ γ E ∗ α − 1 E ∗ β E ] E = [ x 0 , ..., x n , x, y m , ..., y 0 , y 1 , ..., y m , y ] E = [ x 0 , ..., x n , x, y ] E = [ β ] E This implies that [ β ] E ∈ A and finishes the pr o o f that A is uniformly op en. 7 T o prov e 3 ⇒ 1 , note that by Pro pos ition 10, e X has an inv ariant universal basis with resp ect to the isomo r phic action o f δ 1 ( X ). Corolla ry 7 and Lemma 5 together show that φ is a bi-uniformly contin uous surjection, he nce a quo tien t with respect to this actio n (cf. Theorem 11, [6]). Lemma 11 now finishes the pro of. The next corollar y is the statement Coro llary 61 in [3]: Corollary 13 If f : X → Y i s a bi-u niformly c ontinuou s surje ction wher e X is universal and Y is uniform then Y is c over able. Pro of. Accor ding to P rop osition 57 in [3] we hav e the lift f L : X → e Y which satisfies φ ◦ f L = f , where φ : e Y → Y is the pro jection. But then φ m ust be a uniformly contin uous surjectio n. If E is an entourage in e Y , then since f is bi-uniformly contin uous , f ( f − 1 L ( E )) is an ent o ur age that is contained in φ ( E ). This proves that φ is bi-unifor mly contin uous and hence Y is coverable by Theorem 12. Corollary 14 If X is c over able then E is a c overing entour age if and only if X E is chain c onne cte d. Pro of. If E is a covering en tour age then by definition φ E : e X → X E is surjective. Since e X is chain connected, so is X E . Conv erse ly , if X E is chain connected then by Lemma 2 and the thir d part of Theorem 1 2, φ E m ust b e surjective. Note that the argument 3 ⇒ 1 in the pro of of Theorem 12 is constructive; it actually provides a c overing basis. W e can now sort thro ugh the steps to help ident ify this bas is. The pro of of Lemma 11, which is a ctually in [3], shows that the cov er ing entourages are o f the for m φ ( G ), where G is a n inv ariant universal ent o urage in e X . The universal e ntourages in e X come from Pr op osition 10, and they ar e of the form φ − 1 E ( E ∗ ) for any E . Letting A := φ E ( e X ) we have φ ( φ − 1 E ( E ∗ )) = φ ( φ − 1 E ( E ∗ ∩ ( A × A )) = φ X E ◦ φ E ◦ φ − 1 E ( E ∗ ∩ ( A × A )) = E A Note that E A is chain connected since e X is . Combining this with Corollary 3 we obtain: Corollary 15 L et X b e a c over able uniform sp ac e. F or any entour age E , X E has a unique chain c onne cte d uniformly op en set A c ontaining the b asep oint, and E A is a c overing entour age. Example 16 We wil l il lu str ate how Cor ol lary 15 extr acts a c overing entour age fr om a non-c overing ent ou ra ge in the top olo gic al gr oup R . In a top olo gic al gr oup with left uniformity, ent our ages ar e c ompletely determine d by symmetric op en subsets of the identity (which always serves as t he b asep oint). F or example, in R , if U is any such set, t her e c orr esp onds an entour age E ( U ) := { ( x, y ) : x − y ∈ U } . The op en set c orr esp onding t o E ( U ) ∗ in R U := R E ( U ) is denote d by U ∗ . U sing op en sets ra ther than entour ages makes it e asier t o se e what is going on. In 8 Example 48 of [1] we c onsider e d the set U := ( − 1 , 1) ∪ (2 , 4) ∪ ( − 4 , − 2) . In this example t he c omp onents of U ar e far en ou gh ap art t hat R U c onsists of the top olo gic al gr oup R × Z with the pr o duct uniform structure ( Z is discr ete). The ide a her e is that the two outer c omp onents of U c annot b e r e ache d fr om 0 by ( − 1 , 1) -chains, s o for example the e quivalenc e class of the chain { 0 , 3 } lies in a differ ent c omp onent fr om the identity c omp onent A := R × { 0 } in R U . Along these lines, it is not har d to show that U ∗ = ( − 1 , 1) × { 0 } ∪ (2 , 4) × { 1 } ∪ ( − 4 , − 2) × {− 1 } That is, the t wo outer c omp onents of U ∗ do not lie in A , which cle arly is the unique chain c onne cte d uniformly op en set c ontaining the identity in R × Z . Now we have φ R E ( U ) ( U ∗ ∩ A ) = ( − 1 , 1 ) := V . Sinc e V is c onne cte d and R is simply c onne cte d, E ( V ) is a c overing entour age (cf. [1 ]). In [5] the notio n of Rips complex is extended from metric spaces to unifor m spaces: R ( X , E ) is the sub complex of the full complex o ver X having as sim- plices a ll { x 0 , ..., x n } s uc h that ( x i , x j ) ∈ E for all i and j . According to ([8], Section 3.6), a ny path in R ( X , E ) is, up to homoto p y , uniquely identified with a s implicial path, whic h in turn is uniquely determined by its vertices. These vertices, ob vio usly , for m an E - ch a in, and the bas ic mo ves in a fixed-endp oint simplicial homoto p y of simplicial paths (a dding or r emoving a pa ir of edges that span 2-simplex with one edge alr eady in the path) corresp ond pr ecisely to the basic mo ves in a n E - homotopy (adding or removing a p oint so a s to pre- serve that one ha s an E - ch a in). That is, the set o f all fixed-e ndpoint homotopy equiv ale nce clas ses of paths in R ( X , E ) starting at a base point ∗ is naturally ident ified with X E . Using this natural identification of fixed-endp oing homo topies of paths in R ( X , E ) and E -ho motopies of E -chains, we will translate the basic definitions of [5]. Two E -chains α := { x 0 , ..., x n } a nd β := { y 0 , ..., y k } (maybe without the same endpoints) are said in [5] to b e E -homo topic if (1) ( x 0 , y 0 ) , ( x n , y k ) ∈ E and (2 ) β is (fixed-endp oint) E - homotopic to { y 0 , x 0 , ..., x n , y k } . If α and β have the same pair of endp oints then of course “ E -ho motopic” has the same meaning as in [3]. If α and β only have the same starting p oint x 0 = y 0 = ∗ , then it is easy to check that α and β ar e E -homoto pic precisely when ([ α ] E , [ β ] E ) ∈ E ∗ . A gener alize d cur ve fr om x to y is defined in [5] to be a collection { [ c E ] E } of E -homotopy cla sses of E -chains joining x and y such that if F ⊂ E then [ c F ] E = [ c E ] E . The set of all generalized curves sta rting at ∗ is called GP ( X , ∗ ) in [5], but this set is obviously none other than e X via the identification { [ c E ] E } ↔ ([ c E ] E ). The authors define a “natural uniform structure” on GP ( X, ∗ ) (a.k.a. e X ) by taking, for eac h e ntourage F in X , the set of all pa irs ( { [ c E ] E } , { [ d E ] E } ) such that c F is F -homoto pic to d F . Since c F and d F bo th start at ∗ , as p ointed out ab ov e this is equiv ale n t to ([ c F ] F , [ d F ] F ) ∈ F ∗ . That is, the bas is that they define consists precisely o f the sets φ − 1 F ( F ∗ ), whic h of co urse is a bas is for the inv erse limit unifor m structure on e X . In other words, GP ( X , ∗ ) and e X a re one and the s ame space. Moreov er , the mapping π X : GP ( X , ∗ ) → X of [5 ] is 9 the endp oint mapping (identical to φ : e X → X ), and the uniform fundament a l group ∨ π ( X , ∗ ) o f [5] is π − 1 X ( ∗ ) = φ − 1 ( ∗ ) (“ generalized lo ops”) with o per ation induced by conca tenation (identical to δ 1 ( X )). According to [5 ], a uniform space X is called joinable if e v er y pair of p oints in X is joined by a genera lized curve; clearly this is equiv a lent to the s urjectivity of φ : e X → X . In [5] X is called uniformly joinable if for ev er y entourage E there is a n e ntourage F such that whenever ( x, y ) ∈ F , x and y are joined by a generalized curve { [ c D ] D } that is “ E - s hort” in the s ense that [ c E ] E = [ { x, y } ] E . Corollary 17 If X is a cha in c onne cte d uniform sp ac e then X is c over able if and only if X is u n iformly joinable . Pro of. Suppo se that φ E ( e X ) is unifor mly op en for all E ; so there is some F ⊂ E such that if ([ α ] E , [ β ] E ) ∈ F ∗ and [ α ] E ∈ φ E ( e X ) then [ β ] E ∈ φ E ( e X ). Let ( x, y ) ∈ F . Since X is c ha in connected there is so me F - chain α = { x 0 = ∗ , ..., x n − 1 , x } and we may let β := { x 0 , ..., x n − 1 , x, y } . Note that since ([ x 0 , ..., x i ] E , [ x 0 , ..., x i , x i +1 ] E ) ∈ F ∗ for all i , it follows by inductio n on i that [ α ] E ∈ φ E ( e X ). Likewise [ β ] E ∈ φ E ( e X ) and we have [ α ] E = [ α E ] E for some ([ α D ] D ) ∈ e X and [ β ] E = [ β E ] E for some ([ β D ] D ) ∈ e X . But the conc a tenated gener alized cur v e { [ α − 1 D ∗ β D ] D } certainly satisfies the E -short co nditio n [ α − 1 E ∗ β E ] E = [ { x, y } ] E ; in fact one may remov e the p oints x 0 , x 1 , x 1 , ...x n − 1 , x n − 1 , x in succes sion to create an E - homotopy b et ween α − 1 E ∗ β E and { x, y } . If X is uniformly jo inable and E is an entourage, by definition there is so me ent o urage F ⊂ E such that if ( x, y ) ∈ F , x a nd y are joined by a n E -s hort generalized c ur ve. Let α = {∗ = x 0 , ..., x n } b e an E -chain with [ α ] E ∈ φ E ( e X ). If ([ α ] E , [ β ] E ) ∈ F ∗ then by definition of F ∗ we may as sume that β is of the form {∗ = x 0 , ..., x n − 1 , x } with ( x, x n ) ∈ F . That is, there is a n E - short gener alized curve { [ c D ] D } joining x n and x with c E = { x n , x } . No w if φ E ([ α D ] D ) = [ α ] E then g := ([ α D ∗ c D ] D ) ∈ e X satisfie s φ E ( g ) = [ β ] E . Remark 18 In light of Cor ol laries 7 and 17 we have a very nic e way to dis- tinguish b etwe en joinable and uniformly joinable for a chain c onne cte d uniform sp ac e X , namely that for a joinable sp ac e, φ E has uniformly op en image for some E , while for a u niformly joinable sp ac e, φ E has un iformly op en image for al l E . Remark 19 Note that the e qu ivalenc e of The or em 12.2 and uniform joinability for a chain c onn e cte d sp ac e was pr ove d in [5] using c ompletely differ ent ar gu- ments. References [1] V. Berestovskii a nd C . Pla ut, Covering g roup theory for top olo gical g roups, T op. Appl. 114 (20 01) 1 41-186 . 10 [2] V. Berestovskii and C. Plaut, Cov ering gr oup theory for lo cally compac t groups, T op. Appl. 1 14 (2 001) 1 87-19 9. [3] V. Berestovskii and C. Plaut, Uniform Universal Covers of Uniform Spaces, T op. Appl. 154 (20 07) 1 748–17 77. [4] V. Berestovskii and C. Plaut, Cov er ing R -trees, pre print arXiv:070 7.3609 . [5] N. Bro dskiy , J. Dydak, B. Labuz, and A. Mitr a, Rips Complexes and uni- versal cov er s in the unifor m catego ry , preprint arXiv:0 706.39 3 7 . [6] C. Plaut, Quotients o f unifor m spaces, T op. Appl. 1 53 (2 0 06) 243 0-2444 . [7] C. Plaut, Lec tures on Uniform Universal Cov ers of Uniform Spaces (joint work with V. B erestovskii), Universit y o f T ennessee, F eb., 2 007. [8] E . Spa nier, Algebr aic T op olo gy , McGraw-Hill, New Y ork, 19 66. 11